Abstract

Dynamic metasurface antennas are planar structures that exhibit remarkable capabilities in controlling electromagnetic wavefronts, advantages that are particularly attractive for microwave imaging. These antennas exhibit strong frequency dispersion and produce rapidly varying radiation patterns. Such behavior presents unique challenges for integration with conventional imaging algorithms. We adapt the range migration algorithm (RMA) for use with dynamic metasurfaces and propose a preprocessing step that ultimately allows for expression of measurements in the spatial frequency domain, from which the fast Fourier transform can efficiently reconstruct the scene. Numerical studies illustrate imaging performance using conventional methods and the adapted RMA, demonstrating that the RMA can reconstruct images with comparable quality in a fraction of the time. The algorithm can be extended to a broad class of complex antennas for application in synthetic aperture radar and MIMO imaging.

© 2016 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
  38. M. Johnson, P. Bowen, N. Kundtz, and A. Bily, “Discrete-dipole approximation model for control and optimization of a holographic metamaterial antenna,” Appl. Opt. 53, 5791–5799 (2014).
    [Crossref]
  39. L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (4)

T. Fromenteze, E. L. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
[Crossref]

O. Yurduseven, J. N. Gollub, D. L. Marks, and D. R. Smith, “Frequency-diverse microwave imaging using planar mills-cross cavity apertures,” Opt. Express 24, 8907–8925 (2016).
[Crossref]

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” J. Opt. Soc. Am. B 33, 1098–1111 (2016).
[Crossref]

2015 (6)

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” IEEE Trans. Antennas Propag. 63, 593–600 (2015).
[Crossref]

O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Appl. Phys. Lett. 106, 194104 (2015).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107, 204104 (2015).
[Crossref]

T. Sleasman, M. Imani, W. Xu, J. Hunt, T. Driscoll, M. Reynolds, and D. Smith, “Waveguide-fed tunable metamaterial element for dynamic apertures,” IEEE Antennas Wireless Propag. Lett. 15, 606–609 (2015).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
[Crossref]

2014 (4)

C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
[Crossref]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
[Crossref]

M. Johnson, P. Bowen, N. Kundtz, and A. Bily, “Discrete-dipole approximation model for control and optimization of a holographic metamaterial antenna,” Appl. Opt. 53, 5791–5799 (2014).
[Crossref]

2013 (3)

C. Jouvaud, J. de Rosny, and A. Ourir, “Adaptive metamaterial antenna using coupled tunable split-ring resonators,” Electron. Lett. 49, 518–519 (2013).
[Crossref]

H. Odabasi, F. Teixeira, and D. Guney, “Electrically small, complementary electric-field-coupled resonator antennas,” J. Appl. Phys. 113, 084903 (2013).
[Crossref]

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref]

2012 (3)

X. Zhuge and A. G. Yarovoy, “Three-dimensional near-field mimo array imaging using range migration techniques,” IEEE Trans. Image Process. 21, 3026–3033 (2012).
[Crossref]

P. T. Bowen, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14, 033038 (2012).
[Crossref]

C. L. Holloway, E. F. Kuester, J. Gordon, J. O. Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

2011 (1)

S. S. Ahmed, A. Schiessl, and L.-P. Schmidt, “A novel fully electronic active real-time imager based on a planar multistatic sparse array,” IEEE Trans. Microwave Theory Tech. 59, 3567–3576 (2011).
[Crossref]

2009 (2)

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
[Crossref]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[Crossref]

2008 (2)

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93, 212504 (2008).
[Crossref]

M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 46, 1589–1599 (2008).
[Crossref]

2006 (2)

S. Guillaso, A. Reigber, L. Ferro-Famil, and E. Pottier, “Range resolution improvement of airborne sar images,” IEEE Geosci. Remote Sens. Lett. 3, 135–139 (2006).
[Crossref]

I. Gil, J. Bonache, J. García-García, and F. Martín, “Tunable metamaterial transmission lines based on varactor-loaded split-ring resonators,” IEEE Trans. Microwave Theory Tech. 54, 2665–2674 (2006).
[Crossref]

2002 (1)

A. Grbic and G. V. Eleftheriades, “Leaky cpw-based slot antenna arrays for millimeter-wave applications,” IEEE Trans. Antennas Propag. 50, 1494–1504 (2002).
[Crossref]

2000 (1)

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-d radar imaging using range migration techniques,” IEEE Trans Antennas Propag. 48, 728–737 (2000).
[Crossref]

1998 (1)

D. Massonnet and K. L. Feigl, “Radar interferometry and its application to changes in the earth’s surface,” Rev. Geophys. 36, 441–500 (1998).
[Crossref]

1991 (1)

C. Cafforio, C. Prati, and F. Rocca, “Sar data focusing using seismic migration techniques,” Aerosp. Electron. Syst. 27, 194–207 (1991).
[Crossref]

1989 (1)

J.-C. Bolomey, “Recent european developments in active microwave imaging for industrial, scientific, and medical applications,” IEEE Trans. Microwave Theory Tech. 37, 2109–2117 (1989).
[Crossref]

1986 (1)

Y. Saad and M. H. Schultz, “Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
[Crossref]

1984 (1)

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 363–400 (1984).
[Crossref]

1978 (1)

1920 (1)

E. Moors, “On the reciprocal of the general algebraic matrix, abstract,” Bull. Amer. Math. Soc. 26, 394–395 (1920).

Ahmed, S. S.

S. S. Ahmed, A. Schiessl, and L.-P. Schmidt, “A novel fully electronic active real-time imager based on a planar multistatic sparse array,” IEEE Trans. Microwave Theory Tech. 59, 3567–3576 (2011).
[Crossref]

Allan, G.

B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
[Crossref]

Alvarez, Y.

B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
[Crossref]

Ausherman, D. A.

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 363–400 (1984).
[Crossref]

Azad, A. K.

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
[Crossref]

Berkowitz, B.

B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
[Crossref]

Bily, A.

Bily, A. H.

R. A. Stevenson, A. H. Bily, D. Cure, M. Sazegar, and N. Kundtz, “55.2: Invited paper: Rethinking wireless communications: Advanced antenna design using LCD technology,” in SID Symposium Digest of Technical Papers (Wiley, 2015), Vol. 46, pp. 827–830.

Bolomey, J.-C.

J.-C. Bolomey, “Recent european developments in active microwave imaging for industrial, scientific, and medical applications,” IEEE Trans. Microwave Theory Tech. 37, 2109–2117 (1989).
[Crossref]

Bonache, J.

I. Gil, J. Bonache, J. García-García, and F. Martín, “Tunable metamaterial transmission lines based on varactor-loaded split-ring resonators,” IEEE Trans. Microwave Theory Tech. 54, 2665–2674 (2006).
[Crossref]

Booth, J.

C. L. Holloway, E. F. Kuester, J. Gordon, J. O. Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

Bowen, P.

L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
[Crossref]

M. Johnson, P. Bowen, N. Kundtz, and A. Bily, “Discrete-dipole approximation model for control and optimization of a holographic metamaterial antenna,” Appl. Opt. 53, 5791–5799 (2014).
[Crossref]

Bowen, P. T.

P. T. Bowen, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14, 033038 (2012).
[Crossref]

Boyarsky, M.

Brady, D.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref]

Brady, D. J.

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
[Crossref]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[Crossref]

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).

Cafforio, C.

C. Cafforio, C. Prati, and F. Rocca, “Sar data focusing using seismic migration techniques,” Aerosp. Electron. Syst. 27, 194–207 (1991).
[Crossref]

Cannon, T.

Carsenat, D.

T. Fromenteze, E. L. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” IEEE Trans. Antennas Propag. 63, 593–600 (2015).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Appl. Phys. Lett. 106, 194104 (2015).
[Crossref]

Choi, K.

Cummer, S. A.

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93, 212504 (2008).
[Crossref]

Cure, D.

R. A. Stevenson, A. H. Bily, D. Cure, M. Sazegar, and N. Kundtz, “55.2: Invited paper: Rethinking wireless communications: Advanced antenna design using LCD technology,” in SID Symposium Digest of Technical Papers (Wiley, 2015), Vol. 46, pp. 827–830.

Curlander, J. C.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar (Wiley, 1991).

de Rosny, J.

C. Jouvaud, J. de Rosny, and A. Ourir, “Adaptive metamaterial antenna using coupled tunable split-ring resonators,” Electron. Lett. 49, 518–519 (2013).
[Crossref]

Decroze, C.

T. Fromenteze, E. L. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Appl. Phys. Lett. 106, 194104 (2015).
[Crossref]

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” IEEE Trans. Antennas Propag. 63, 593–600 (2015).
[Crossref]

Dehmollaian, M.

M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 46, 1589–1599 (2008).
[Crossref]

Dienstfrey, A.

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
[Crossref]

Driscoll, T.

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J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
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P. T. Bowen, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14, 033038 (2012).
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T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Appl. Phys. Lett. 106, 194104 (2015).
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T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Appl. Phys. Lett. 106, 194104 (2015).
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O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
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G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
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Grbic, A.

A. Grbic and G. V. Eleftheriades, “Leaky cpw-based slot antenna arrays for millimeter-wave applications,” IEEE Trans. Antennas Propag. 50, 1494–1504 (2002).
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S. Guillaso, A. Reigber, L. Ferro-Famil, and E. Pottier, “Range resolution improvement of airborne sar images,” IEEE Geosci. Remote Sens. Lett. 3, 135–139 (2006).
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H. Odabasi, F. Teixeira, and D. Guney, “Electrically small, complementary electric-field-coupled resonator antennas,” J. Appl. Phys. 113, 084903 (2013).
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T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93, 212504 (2008).
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C. L. Holloway, E. F. Kuester, J. Gordon, J. O. Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
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[Crossref]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
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J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
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L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
[Crossref]

T. Sleasman, M. Imani, W. Xu, J. Hunt, T. Driscoll, M. Reynolds, and D. Smith, “Waveguide-fed tunable metamaterial element for dynamic apertures,” IEEE Antennas Wireless Propag. Lett. 15, 606–609 (2015).
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T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” J. Opt. Soc. Am. B 33, 1098–1111 (2016).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
[Crossref]

O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
[Crossref]

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T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107, 204104 (2015).
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T. Fromenteze, E. L. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
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C. L. Holloway, E. F. Kuester, J. Gordon, J. O. Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
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Kundtz, N. B.

P. T. Bowen, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14, 033038 (2012).
[Crossref]

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B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
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Lipworth, G.

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
[Crossref]

O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
[Crossref]

C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
[Crossref]

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref]

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J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-d radar imaging using range migration techniques,” IEEE Trans Antennas Propag. 48, 728–737 (2000).
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B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
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Martín, F.

I. Gil, J. Bonache, J. García-García, and F. Martín, “Tunable metamaterial transmission lines based on varactor-loaded split-ring resonators,” IEEE Trans. Microwave Theory Tech. 54, 2665–2674 (2006).
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B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
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J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
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B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
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C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
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O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
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G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
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A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989), Vol. 2.

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C. Jouvaud, J. de Rosny, and A. Ourir, “Adaptive metamaterial antenna using coupled tunable split-ring resonators,” Electron. Lett. 49, 518–519 (2013).
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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
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D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 363–400 (1984).
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B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
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T. Sleasman, M. Imani, W. Xu, J. Hunt, T. Driscoll, M. Reynolds, and D. Smith, “Waveguide-fed tunable metamaterial element for dynamic apertures,” IEEE Antennas Wireless Propag. Lett. 15, 606–609 (2015).
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J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
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B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
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G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
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T. Fromenteze, E. L. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
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T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” J. Opt. Soc. Am. B 33, 1098–1111 (2016).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107, 204104 (2015).
[Crossref]

T. Sleasman, M. Imani, W. Xu, J. Hunt, T. Driscoll, M. Reynolds, and D. Smith, “Waveguide-fed tunable metamaterial element for dynamic apertures,” IEEE Antennas Wireless Propag. Lett. 15, 606–609 (2015).
[Crossref]

C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

Smith, D.

L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
[Crossref]

T. Sleasman, M. Imani, W. Xu, J. Hunt, T. Driscoll, M. Reynolds, and D. Smith, “Waveguide-fed tunable metamaterial element for dynamic apertures,” IEEE Antennas Wireless Propag. Lett. 15, 606–609 (2015).
[Crossref]

Smith, D. R.

O. Yurduseven, J. N. Gollub, D. L. Marks, and D. R. Smith, “Frequency-diverse microwave imaging using planar mills-cross cavity apertures,” Opt. Express 24, 8907–8925 (2016).
[Crossref]

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” J. Opt. Soc. Am. B 33, 1098–1111 (2016).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
[Crossref]

O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Appl. Phys. Lett. 106, 194104 (2015).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107, 204104 (2015).
[Crossref]

C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
[Crossref]

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref]

P. T. Bowen, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14, 033038 (2012).
[Crossref]

C. L. Holloway, E. F. Kuester, J. Gordon, J. O. Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93, 212504 (2008).
[Crossref]

Soumekh, M.

M. Soumekh, Synthetic Aperture Radar Signal Processing (Wiley, 1999).

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R. A. Stevenson, A. H. Bily, D. Cure, M. Sazegar, and N. Kundtz, “55.2: Invited paper: Rethinking wireless communications: Advanced antenna design using LCD technology,” in SID Symposium Digest of Technical Papers (Wiley, 2015), Vol. 46, pp. 827–830.

Taylor, A. J.

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
[Crossref]

Teixeira, F.

H. Odabasi, F. Teixeira, and D. Guney, “Electrically small, complementary electric-field-coupled resonator antennas,” J. Appl. Phys. 113, 084903 (2013).
[Crossref]

Trofatter, P.

Valayil, M.

L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
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[Crossref]

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C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

Xu, W.

T. Sleasman, M. Imani, W. Xu, J. Hunt, T. Driscoll, M. Reynolds, and D. Smith, “Waveguide-fed tunable metamaterial element for dynamic apertures,” IEEE Antennas Wireless Propag. Lett. 15, 606–609 (2015).
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X. Zhuge and A. G. Yarovoy, “Three-dimensional near-field mimo array imaging using range migration techniques,” IEEE Trans. Image Process. 21, 3026–3033 (2012).
[Crossref]

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O. Yurduseven, J. N. Gollub, D. L. Marks, and D. R. Smith, “Frequency-diverse microwave imaging using planar mills-cross cavity apertures,” Opt. Express 24, 8907–8925 (2016).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54, 9343–9353 (2015).
[Crossref]

O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Appl. Phys. Lett. 106, 194104 (2015).
[Crossref]

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X. Zhuge and A. G. Yarovoy, “Three-dimensional near-field mimo array imaging using range migration techniques,” IEEE Trans. Image Process. 21, 3026–3033 (2012).
[Crossref]

Zvolensky, T.

L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
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[Crossref]

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93, 212504 (2008).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107, 204104 (2015).
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IEEE Antennas Propag. Mag. (1)

C. L. Holloway, E. F. Kuester, J. Gordon, J. O. Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (2)

T. Sleasman, M. Imani, W. Xu, J. Hunt, T. Driscoll, M. Reynolds, and D. Smith, “Waveguide-fed tunable metamaterial element for dynamic apertures,” IEEE Antennas Wireless Propag. Lett. 15, 606–609 (2015).
[Crossref]

L. Pulido-Mancera, T. Zvolensky, M. Imani, P. Bowen, M. Valayil, and D. Smith, “Discrete dipole approximation applied to highly directive slotted waveguide antennas,” IEEE Antennas Wireless Propag. Lett. PP, 1 (2016).
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IEEE Trans. Aerosp. Electron. Syst. (1)

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 363–400 (1984).
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IEEE Trans. Antennas Propag. (3)

B. Gonzalez-Valdes, G. Allan, Y. Rodriguez-Vaqueiro, Y. Alvarez, S. Mantzavinos, M. Nickerson, B. Berkowitz, J. A. Martinez-Lorenzo, F. Las-Heras, and C. M. Rappaport, “Sparse array optimization using simulated annealing and compressed sensing for near-field millimeter wave imaging,” IEEE Trans. Antennas Propag. 62, 1716–1722 (2014).
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H. Odabasi, F. Teixeira, and D. Guney, “Electrically small, complementary electric-field-coupled resonator antennas,” J. Appl. Phys. 113, 084903 (2013).
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J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am A 31, 2109–2119 (2014).
[Crossref]

J. Opt. Soc. Am. B (1)

Metamaterials (1)

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
[Crossref]

Nat. Photonics (1)

C. M. Watts, D. Shrekenhamer, J. Montoya, G. Lipworth, J. Hunt, T. Sleasman, S. Krishna, D. R. Smith, and W. J. Padilla, “Terahertz compressive imaging with metamaterial spatial light modulators,” Nat. Photonics 8, 605–609 (2014).
[Crossref]

New J. Phys. (1)

P. T. Bowen, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14, 033038 (2012).
[Crossref]

Opt. Express (2)

Prog. Electromag. Res. (1)

O. Yurduseven, M. F. Imani, H. Odabasi, J. Gollub, G. Lipworth, A. Rose, and D. R. Smith, “Resolution of the frequency diverse metamaterial aperture imager,” Prog. Electromag. Res. 150, 97–107 (2015).
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R. A. Stevenson, A. H. Bily, D. Cure, M. Sazegar, and N. Kundtz, “55.2: Invited paper: Rethinking wireless communications: Advanced antenna design using LCD technology,” in SID Symposium Digest of Technical Papers (Wiley, 2015), Vol. 46, pp. 827–830.

M. Soumekh, Synthetic Aperture Radar Signal Processing (Wiley, 1999).

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).

M. Soumekh, Fourier Array Imaging (Prentice-Hall, 1994).

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, 1989), Vol. 2.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar (Wiley, 1991).

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Figures (10)

Fig. 1.
Fig. 1.

Dynamic metasurface antenna.

Fig. 2.
Fig. 2.

(a) Bistatic SAR imaging system. (b) Dynamic metasurface antenna imaging system.

Fig. 3.
Fig. 3.

Block diagram of the adapted RMA applied for a stationary dynamic metasurface.

Fig. 4.
Fig. 4.

Singular value spectra for different sets of masks applied. (a)  Φ for identity masks. (b)  Φ for random masks. (c) SV spectrum.

Fig. 5.
Fig. 5.

Transformation of the signal measured with the dynamic metasurface to apply the RMA. (a)  S n y × n f , (b)  g n m × n f , (c)  S ^ n y × n f .

Fig. 6.
Fig. 6.

Steps of the range migration algorithm for a PSF. (a)  S ^ n y × n f . (b) FFT in cross range. (c) Stolt interpolation. (d) 2D-IFFT.

Fig. 7.
Fig. 7.

(a) 2D map of the PSNR as a function of the regularization parameter and the oversampling rate nm/ny. (b) Truncated SVD curve for three different scenarios. (c) Image reconstruction for each scenario: 1. PSNR = 26.5    dB ; 2. PSNR = 33.2    dB , 3. PSNR = 23.8    dB .

Fig. 8.
Fig. 8.

Cross sections of the PSF in (a) range and (b) cross range. Different colors correspond to different reconstruction methods.

Fig. 9.
Fig. 9.

Image reconstruction of a complex target with different reconstruction methods. (a) Object. (b) Matched filter. (c) GMRES. (d) RMA.

Fig. 10.
Fig. 10.

3D image reconstruction by moving the dynamic metasurface antenna along the z direction. (a) Target locations. (b) Reconstructed images.

Tables (1)

Tables Icon

Table 1. Precomputing Time (PT) and Reconstruction Time (RT) Using Different Techniques a

Equations (42)

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H g ( y ) = H 0 e j β y ,
α ( y i ) = F ω 2 ω 2 ω 0 ( y i ) 2 + j ω γ ( y i ) .
m ( y i ) = α ( y i ) H g ( y i ) = α ( y i ) H 0 e j β y i .
U ( r⃗ ; f ) i Z 0 k ω m ( y i ) 4 π | r⃗ y i y ^ | e j k | r⃗ y i y ^ | sin θ ,
U ( r⃗ ; f ) Z 0 k ω 4 π | r⃗ | i α ( y i ) e j β y i e j k y i sin θ .
S ( y t , y r ; f ) = V G ( y t , r⃗ ; f ) σ ( r⃗ ) G ( y r , r⃗ ; f ) d V ,
G ( r⃗ , r⃗ ; f ) = e j 2 π f c | r⃗ r⃗ | | r⃗ r⃗ | .
g ( f ) = V U t ( r⃗ ; f ) σ ( r⃗ ) U r ( r⃗ ; f ) d V .
g ( f ) = V H ( r⃗ ; f ) σ ( r⃗ ) d V ,
g = H σ + n ,
U m ( y i ; f ) Z 0 H 0 α m ( y i ; f ) e j β ( f ) y i .
S ( y i ; f ) = m g m ( f ) U m ( y i ; f ) = m Φ m ( y i ; f ) g m ( f ) ,
Φ m ( y i ; f ) Z 0 H 0 α m ( y i ; f ) e j β ( f ) y i .
i Φ m ( y i ; f ) Φ m + ( y i ; f ) = δ m , m | Z 0 H 0 | 2 ,
i Φ m + ( y i ; f ) S ( y i ; f ) = i m g m ( f ) Φ m ( y i ; f ) Φ m + ( y i ; f ) .
g m = i Φ m + ( y i ; f ) S ( y i ; f ) .
Φ g = S ,
Φ Φ + = I .
S ( y t , f ) = V σ ( x , y , z ) 16 π 2 R t R r e j k R t e j k R r δ ( y r ) δ ( z r ) δ ( z t ) d V .
R t = x 2 + ( y y t ) 2 + ( z z t ) 2 .
R r = x 2 + ( y y r ) 2 + ( z z r ) 2 .
S ( k y , k ) = A t A r S ( y t , k ) d A t d A r = V σ ( x , y , z ) 16 π 2 A t e j k R t R t e j k y t y t e j k z t z t δ ( z t ) d A t × A r e j k R r R r e j k y r y r e j k z r z r δ ( z r ) δ ( y r ) d A r d V .
A r e j k R r R r e j k y r y r e j k z r z r δ ( z r ) δ ( y r ) d A r = e j k R R ,
A t e j k R t R t e j k y t y t e j k z t z t δ ( z t ) d A t = y t e j k R R e j k y t y t d y t ,
y t e j k R R e j k y t y t d y t = K E 1 ( k y t ) E 2 ( k y t ) e j k y t y d k y t ,
E 1 ( k y t ) = y e j k R R e j k y t y d y j 2 π k 2 k y t 2 e j k 2 k y t 2 x 2 + z 2 .
E 2 ( k y t ) = y e j k y t y e j k y t y d y = δ ( k y t + k y t ) .
y t e j k R t e j k y t y t d y t j 2 π k 2 k y t 2 e j k 2 k y t 2 x 2 + z 2 e j k y t y .
S ( k y , k ) = V σ ( x , y , z ) 16 π 2 j 2 π k 2 k y t 2 × e j k x 2 + y 2 + z 2 e j k 2 k y t 2 x 2 + z 2 e j k y t y d V .
S ( k y , k x ) = j 2 π 16 π 2 x y σ ( x , y , z ) k 2 k y 2 e j k x x e j k y y d y d x ,
k y = k y t k x = k 2 k y 2 + k .
S ( z t , y t , f ) = j 2 π 16 π 2 V σ ( x , y , z ) R t R r e j k R t e j k R r δ ( y r ) δ ( z r z t ) d V .
S ( k z , k y , k x ) = x y z j 2 π σ ( x , y , z ) 16 π 2 k 2 k y 2 k z 2 × e j x k 2 k y 2 k z 2 / 2 e j x 2 + y 2 k 2 k y 2 k z 2 / 2 e j k y y e j k z z d z d y d x ,
S ( k z , k y , k x ) = x y z j 2 π σ ( x , y , z ) 16 π 2 k 2 k y 2 k z 2 e j k x x e j k y y e j k z z d z d y d x ,
k x = k 2 k y 2 k z 2 / 2 + k 2 k z 2 / 2 .
Φ = U Σ V ,
Φ + = U Σ + V ,
Σ + = [ 1 s 1 0 0 1 s n y ] .
PSNR = 10 log 10 ( m = 1 M n = 1 N | I S ( m , n ) I S ^ ( m , n ) | M N ) .
σ est = H g ,
δ r = c 2 B ,
δ c r = λ c x 0 L ,

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